Social Security benefits become taxable if your "combined income" exceeds $25,000 (single) or $32,000 (married filing jointly). Up to 50% of benefits are taxed if you exceed these amounts; up to 85% is taxed if income exceeds $34,000 (single) or $44,000 (joint).
You have the option of having tax withheld from your social security and your part time job but I would suggest submitting a new W4 to your employer to increase the amount of tax withheld from your pay to cover the increased amount of tax due to social security. That way if you do not continue your part time job at some point, you will not need to readjust your social security. At that point, if you combined income decreases, less of your social security will be taxed.
Here is the link to the TurboTax W4 calculator which will help you calculate how much tax you should have withheld and you can print out a new W4 to give your employer.
https://turbotax.intuit.com/tax-tools/calculators/w4/
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